Let $Z = \mathbb{F}_q = \set{z_1, \ldots, z_q}$ be a subfield
of $D$. Taking advantage of the fact that $Z$ is then cyclic,
and the center of $D$, we may write any element in $D$ as
\[ z_1 d_{11}d_{12} \cdots d_{1n_1} + z_2 d_{21} d_{22} \cdots
d_{2n_2} + \cdots + z_q d_{q1} d_{q2} \cdots d_{q_n}, \]
where $d_{jk} \notin Z$.